Asymptotic analysis of average case approximation complexity of additive random fields

Abstract

We study approximation properties of sequences of centered additive random fields $Y_d$, $d\in \mathbb{N}$. The average case approximation complexity $n^{Y_d}(\epsilon )$ is defined as the minimal number of evaluations of arbitrary linear functionals that is needed to approximate $Y_d$ with relative $2$-average error not exceeding a given threshold $\epsilon \in (0,1)$. We investigate the growth of $n^{Y_d}(\epsilon )$ for arbitrary fixed $\epsilon \in (0,1)$ and $d\to \infty$. Under natural assumptions we obtain general results concerning asymptotics of $n^{Y_d}(\epsilon )$. We apply our results to additive random fields with marginal random processes corresponding to the Korobov kernels.